Volume 21, 2015, Number 4

This paper is concerned with integer solutions of the Diophantine equation x14 + x24 + x34 = k x42 where k is a given positive integer. Till now, integer and parametric solutions of this Diophantine equation have been published only when k = 1 or 2 or 3. In this paper we obtain parametric solutions of this equation for 43 values of k ≤ 100. We also show that the equation cannot have any solution in integers for 54 values of k ≤ 100. The solvability of the equation x14 + x24 + x34 = k x42 where k could not be determined for three values of k ≤ 100, namely 34, 35 and 65.

In this paper we consider Diophantine triples {a, b, c} (denoted D(n)-3-tuples) and give necessary and sufficient conditions for existence of integer n given the 3-tuple {a, b, c} so that ab + n, ac + n, bc + n are all squares of integers. Several examples as applications of the main results, related to both Diophantine triples and quadruples, are given.

Only prime integers that are in Class ̅14 of the Modular Ring Z4 equate to a sum of squares of integers x and y. A simple equation to predict these integers is developed which distinguishes prime and composite numbers in that one (x, y) couple exists for primes, but composites have either one couple with a common factor or the same number of couples as there are factors. In particular, composite Fibonacci numbers always have multiple (x, y) couples because the factors are all elements of ̅14.

Two triangular number based primality tests for numbers in the arithmetic progressions 8n ± 1 are obtained. Their use yield a new Diophantine approach to the existence of an infinite number of twin primes of the form (8n−1, 8n+1).

We revisit Elder’s theorem on integer partitions, which is a generalization of Stanley’s theorem. Two new proofs are presented. The first proof is based on certain tilings of 1 × ∞ boards while the second one is a consequence of a more general identity we prove using generating functions.

We consider certain properties of functions f : J → I (I, J intervals) such that f(M(x, y)) ≤ N(f(x), f(y)), where M and N are general means. Some results are extensions of the case M = N = L, where L is the logarithmic mean.

Consider the multiplicative group of integers modulo n, denoted by ℤ*n. An element a ∈ ℤ*n is said to be a semi-primitive root modulo n if the order of a is φ (n)/2, where φ(n) is the Euler’s phi-function. In this paper, we’ll discuss on the number of semi-primitive roots of non-cyclic group ℤ*n and study the relation between S(n) and K(n), where S(n) is the set of all semi-primitive roots of non-cyclic group ℤ*n and K(n) is the set of all quadratic non-residues modulo n.

The structure of the ‘Golden Ratio Family’ is consistent enough to permit the primality tests developed for φ5 to be applicable. Moreover, the factors of the composite numbers formed by a prime subscripted member of the sequence adhere to the same pattern as for φ5. Only restricted modular class structures allow prime subscripted members of the sequence to be a sum of squares. Furthermore, other properties of φ5 are found to apply to those other members with structural compatibility.

In this note, we provide a combinatorial proof of a generalized recurrence formula satisfied by the Stirling numbers of the second kind. We obtain two extensions of this formula, one in terms of r-Whitney numbers and another in terms of q-Stirling numbers of Carlitz. Modifying our proof yields analogous formulas satisfied by the r-Stirling numbers of the first kind and by the r-Lah numbers.